0,2

a(n) = Sum_{k=0..n} binomial(n,k)*A057977(k). For comparison:

A001006(n) = Sum_{k=0..n} binomial(n,k)*A057977(k)*[k is even],

A005717(n) = Sum_{k=0..n} binomial(n,k)*A057977(k)*[k is odd].

Thus one might simply say: The extended Motzkin numbers are the binomial sum of the extended Catalan numbers. Moreover: The Catalan numbers aerated with 0's at odd positions (A126120) are the inverse binomial transform of the Motzkin numbers (A001006). The complementary Catalan numbers (A001700) aerated with 0's at even positions (A138364) are the inverse binomial transform of the complementary Motzkin numbers (A005717). The extended Catalan numbers (A057977 = A126120 + A138364) are the inverse binomial transform of the extended Motzkin numbers (A189912).

David Scambler observed that [1, a(n-1)] for n >= 1 count the Dyck paths of semilength n which satisfy the condition "number of peaks <= number of returns + number of hills". - Peter Luschny, Oct 22 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000

Per Alexandersson, Proof of Werner Schulte's formula.

A. Asinowski and G. Rote, Point sets with many non-crossing matchings, arXiv preprint arXiv:1502.04925 [cs.CG], 2015.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Peter Luschny, The Scambler_statistic_on_Dyck_words.

a(n) = Sum_{k=0..n} n!/(((n-k)!*floor(k/2)!^2)*(floor(k/2)+1)).

Recurrence: (n+2)*(n^2 + 2*n - 5)*a(n) = (2*n^3 + 7*n^2 - 14*n - 7)*a(n-1) + 3*(n-1)*(n^2 + 4*n - 2)*a(n-2). - Vaclav Kotesovec, Mar 20 2014

a(n) ~ 3^(n+1/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014

Conjecture: a(n) = Sum_{k=0..floor(n/2)} (n+1-2*k)*A055151(n,k). - Werner Schulte, Oct 23 2016

a(n) = Sum_{k=0..floor(n/2)} (n+1-2*k)*n!/(k!*(k+1)!*(n-2*k)!). - Per W. Alexandersson, May 28 2020

A189912 := proc(n) local k;

add(n!/(((n-k)!*iquo(k, 2)!^2)*(iquo(k, 2)+1)), k=0..n) end:

M := proc(n) option remember; `if`(n<2, 1, (3*(n-1)*M(n-2)+(2*n+1)*M(n-1))/(n+2)) end:

A189912 := n -> n*M(n-1)+M(n);

seq(A189912(i), i=0..28); # Peter Luschny, Sep 12 2011

A057977[n_] := n!/(Quotient[n, 2]!^2*(Quotient[n, 2] + 1)); a[n_] := Sum[Binomial[n, k]*A057977[k], {k, 0, n}]; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, May 21 2013, after Peter Luschny *)

Table[Sum[n!/(((n-k)!*Floor[k/2]!^2)*(Floor[k/2]+1)), {k, 0, n}], {n, 0, 30}] (* G. C. Greubel, Jan 24 2017 *)

A057977[n_] := Sum[n! (n + 1 - 2 k)/((k + 1)! (k!) (n - 2 k)!), {k, 0, n}] (* Per W. Alexandersson, May 28 2020 *)

(SageMath)

@CachedFunction

def M(n): return (3*(n-1)*M(n-2)+(2*n+1)*M(n-1))/(n+2) if n>1 else 1

A189912 = lambda n: n*M(n-1) + M(n)

[A189912(i) for i in (0..28)] # Peter Luschny, Oct 22 2012

(PARI) a(n) = sum(k=0, n, binomial(n, k)*k!/( (k\2)!^2 * (k\2+1)) );

vector(30, n, a(n-1)) \\ G. C. Greubel, Jan 24 2017; Mar 28 2020

Cf. A001006, A001700, A005717, A057977, A126120, A138364, A217539, A217540.

Sequence in context: A166516 A230552 A230555 * A268321 A195981 A124500

Adjacent sequences: A189909 A189910 A189911 * A189913 A189914 A189915

Peter Luschny, May 01 2011

approved